Some basic properties and fundamental relations for discrete Muckenhoupt and Gehring classes

Abstract

In this paper, we prove some basic properties of the discrete Muckenhoupt class $\mathcal{A}^{p}$ A p and the discrete Gehring class $\mathcal{G}^{q}$ G q . These properties involve the self-improving properties and the fundamental transitions and inclusions relations between the two classes.

Theory of discrete Muckenhoupt weights and discrete Rubio de Francia extrapolation theorems

“…These results further illustrate the value of the extrapolation of families of pairs of functions. To prove these results we need the following inclusion properties of Muckenhoupt classes that has been proved in [28].…”

“…Lemma 1.1. [28] Assume that w is a nonnegative weight and 1 < p, q < ∞ are positive real numbers. Then the following properties hold.…”

“…In the last years, there has been renewed interest in the area of discrete harmonic analysis and then it becomes an active field of research. For example, the study of discrete ℓ p (w) analogues for L p (w) results has been considered by some authors, see for example [5,6,7,8,17,23,24,25,28] and the references cited therein. In [4] the authors mentioned that the study of discrete norm inequalities is not an easy task and more difficult to analyze than its integral counterparts and discovered that the discrete conditions do not correspond, in any natural way, with those that are obtained by discretization the results of functions but the reverse is true.…”

Properties of Muckenhoupt and Gehring classes via conformable calculus

“…For example, the research on regularity and boundedness of discrete analogues of the corresponding L p operators, as well as the higher summability theorems analogues for higher integrability theorems have been studied by numerous authors (see, e.g., [6,8,16,17,24,28] and references therein). In the articles [23,26] the authors studied the structure and basic properties of the weighted discrete Gehring classes, as well as the relationship between the weighted discrete Gehring and Muckenhoupt classes. In recent years, by utilizing the conformable calculus, many authors proved several results related to some integral inequalities like Chebyshev type inequality [3], Hardy type inequalities [25], Hermite-Hadamard type inequalities [2,12,13], and Iyengar type inequalities [27].…”

On discrete weighted Hardy type inequalities and properties of weighted discrete Muckenhoupt classes

“…In [32], the authors proved that v ∈ A p , 1 < p < ∞ if and only if v -1 p-1 ∈ A p , where p is the conjugate of p, and if v ∈ A p (1 < p < ∞), then v ∈ A q for every q > p and that v α ∈ A p for any 0 < α < 1. Böttcher and Seybold [4] proved that if v ∈ A p then there is ε = ε p,v such that v r ∈ A p for all r ∈ (1ε, 1 + ε).…”